The average of the five numbers 26, 29, n, 35 and 43 lies between 25 and 35. If n is an integer and must be greater than the average of these five numbers, then what is the smallest possible value of n?

Introduction / Context:
This question combines inequalities with the concept of average and an additional condition on one of the numbers. We are given four fixed numbers and a variable integer n, and told that the average of the five numbers lies between 25 and 35. We are also told that n itself must be greater than this average. We must find the smallest integer n that can satisfy both conditions simultaneously.

Given Data / Assumptions:

• The five numbers are 26, 29, n, 35 and 43.
• Their average lies strictly between 25 and 35.
• n is an integer.
• n must be greater than the average of the five numbers.
• We are asked for the smallest such integer value of n.

Concept / Approach:
The average of the five numbers is (26 + 29 + n + 35 + 43) / 5. First, we enforce the condition that this average lies between 25 and 35, which gives an inequality for n. Next, we apply the condition that n is greater than this average, producing another inequality. Combining these inequalities and then restricting to integer values gives a range of possible n. Finally, we choose the smallest integer in that range.

Step-by-Step Solution:
Step 1: Compute the sum of the four fixed numbers: 26 + 29 + 35 + 43 = 133. Step 2: The average of the five numbers is (133 + n) / 5. Step 3: The average lies between 25 and 35: 25 < (133 + n) / 5 < 35. Step 4: Multiply through by 5: 125 < 133 + n < 175. Step 5: Subtract 133: -8 < n < 42. So from this condition alone, n can be any number less than 42 and greater than -8. Step 6: Now use the condition that n is greater than the average: n > (133 + n) / 5. Step 7: Multiply both sides by 5: 5n > 133 + n. Step 8: Subtract n from both sides: 4n > 133. Step 9: Divide by 4: n > 133 / 4 = 33.25. Step 10: Combining both sets of inequalities gives 33.25 < n < 42. Step 11: Since n must be an integer, possible values are 34, 35, 36, 37, 38, 39, 40 and 41. Step 12: We are asked for the smallest possible integer n, so n = 34.

Verification / Alternative check:
Check n = 34. The average is (133 + 34) / 5 = 167 / 5 = 33.4. This is between 25 and 35, and n = 34 is indeed greater than 33.4. If we try n = 33, the average is (133 + 33) / 5 = 166 / 5 = 33.2; although this lies within 25 and 35, n = 33 would not be greater than the average, because 33 is slightly less than 33.2. Thus, n = 34 is the smallest integer that satisfies both conditions.

Why Other Options Are Wrong:
Option 33 fails the condition n > average, as shown above. The value 35 does satisfy the conditions but is not the smallest such integer. The option 'none of these' is incorrect because 34 works perfectly. Any larger value like 36 also works but is not minimal. Therefore, 34 is the only correct choice given the wording of the problem and the desire for the smallest possible n.

Common Pitfalls:
Some students only apply the first inequality (average between 25 and 35) and choose any value in that range, without enforcing the second condition that n must be greater than the average. Others may misinterpret 'greater than the average' as greater than some rough mental estimate and not check carefully. It is also easy to forget that n must be an integer, which is crucial when dealing with strict inequalities and averages.

Final Answer:
The smallest integer value of n that satisfies all conditions is 34.